arXiv subjects

Yannic Steenbeck

Publications and source records attributed to Yannic Steenbeck.

9 recordsLinked to original sources

arXiv2026

Quenched functional central limit theorem for the random conductance model under minimal moments

We prove a quenched functional central limit theorem for the random conductance model with ergodic translation-invariant, strictly positive nearest-neighbor conductances, assuming only finite first moments of the conductances and their inverses. This settles an open problem by reaching the critical first-moment threshold, which is sharp for a certain class of integrability assumptions. A key ingredient of the proof is a new Sobolev inequality that seems to be missing from the literature.

math.PR↗

arXiv2026

On the stationary measures of the critical Ornstein--Uhlenbeck process

We study linear and symmetric diffusion processes on $\mathbb{R}^{\mathbb{Z}^d}$ that can be seen as the Langevin dynamics of the (inhomogeneous) harmonic crystal for given conductances, with a particular focus on their stationary distributions. Despite the linearity of the interaction, we exhibit a rich variety of behaviours, depending on the disorder encoded by the conductances. Our main results provide essentially sharp criteria on the conductances ensuring that every stationary measure is reversible. We also provide examples for which these criteria fail and where either no stationary measures exist or reversible and non-reversible stationary measures coexist. Additionally, we show that the linear system can even exhibit non-trivial time-periodic behaviour and provide a spectral characterisation of the occurrence of such oscillations. The case of deterministic conductances is complemented by a study of the case of random conductances under quite general moment assumptions.

math.PR↗

arXiv2026

Can one feel the existence of a non-trivial invariant measure?

It is shown that a bounded linear map on a complex separable Hilbert space with non-trivial invariant probability measures doesn't have to possess eigenvalues. This resolves a question indicated by Flytzanis in 1995 and concretely asked by Grivaux--López-Martínez from 2023 resp. Grivaux--Matheron--Menet from 2021. Still, as a positive result, we prove that every bounded linear operator on a separable complex Banach space for which a non-fixing invariant probability measure exists, has to have some approximate point spectrum on the unit circle minus $1$.

math.FA↗

arXiv2026

Asymptotically complete free-energy dissipation: a coarse MLSI holds at any positive temperature

Everybody learns in school that an out-of-equilibrium system coupled to a heat bath at a fixed temperature evolves to thermodynamic equilibrium as time goes on, and the free energy will only decrease on its way there. At least since Holley's 1971 work, mathematicians know this too, in the modest context of classical Ising Glauber dynamics. But does the free energy also asymptotically decrease to the free energy of an equilibrium state? A typical school kid would say "yes, of course", but since the free energy is only lower semicontinuous, this question is less straightforward than it initially seems. To the best of our knowledge, apart from the uniqueness regime, where one can make use of classical functional inequalities, this question has not previously been resolved rigorously. We prove the asymptotically complete dissipation of the free energy for the classical Ising Glauber dynamics by introducing a coarse modified log-Sobolev inequality, which holds at every positive temperature, in particular in the phase-coexistence regime.

math.PR↗

arXiv2026

Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures

We prove that there is only one translation-invariant Gibbsian point process w.r.t. to a chosen interaction if any of them satisfies a certain bound related to concentration-of-measure. This concentration-of-measure bound is e.g. fulfilled if a corresponding modified logarithmic Sobolev inequality holds. In particular, for natural examples with non-uniqueness regimes, a modified logarithmic Sobolev inequality cannot be satisfied. Therefore, in these situations, the free-energy dissipation in related continuous-time birth-and-death dynamics in $\mathbb{R}^d$ is not exponentially fast.

math.PR↗

arXiv2026

Reversible birth-and-death dynamics in continuum: a de Bruijn-type identity for free-energy dissipation

We investigate free-energy dissipation in a continuous-time birth-and-death dynamics in $\mathbb{R}^d$. For these Markov processes, the class of reversible measures coincides with the infinite-volume Gibbs point processes for some sufficiently nice Hamiltonian. For a wide class of initial distributions, we derive a de~Bruijn-type identity that relates the time evolution of the specific relative entropy along trajectories to the Fisher information, in particular establishing the thermodynamic limit of the latter. Along the way, we analyze some fine properties of the considered dynamics, such as the existence and regularity of local densities, obtain a spatial ergodic theorem for the entropy production per unit volume, and derive a small-time exponential series expansion of the dynamics.

math.PR↗

arXiv2025

The variational principle for a marked Gibbs point process with infinite-range multibody interactions

We prove the Gibbs variational principle for the Asakura--Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a $k$-body interaction for fixed $k$. As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.

math.PR↗

arXiv2025

Reversible birth-and-death dynamics in continuum: free-energy dissipation and attractor properties

We consider continuous-time birth-and-death dynamics in $\mathbb{R}^d$ that admit at least one infinite-volume Gibbs point process based on area interactions as a reversible measure. For a large class of starting measures, we show that the specific relative entropy decays along trajectories, and that all possible long-time weak limit points are also Gibbs point processes with respect to the same interaction. Our proof rests on a representation of the entropy dissipation in terms of the Palm version of the propagated measure.

math.PR↗

arXiv2025

Throughput in inhomogeneous planar drainage networks

We consider navigation schemes on planar diluted lattices and semi lattices with one discrete and one continuous component. More precisely, nodes that survive inhomogeneous Bernoulli site percolation, or are placed as inhomogeneous Poisson points on shifted copies of $\mathbb{Z}$, forward their individually generated traffic to their respective closest neighbors to the left in the next layer. The resulting drainage network is a tree and we study the amount of traffic that goes through an increasing window at the origin. Our main results show that, properly rescaled, the total traffic, jointly with the total length of the contributing tree part, converges to the area under a time-inhomogeneous Brownian motion until it hits zero. The hitting time corresponds to the limiting maximal path length.

math.PR↗